Let $ABCD$ be a right-angled trapezoid and $M$ be the midpoint of its greater lateral side $CD$. Circumcircles $\omega_{1}$ and $\omega_{2}$ of triangles $BCM$ and $AMD$ meet for the second time at point $E$. Let $ED$ meet $\omega_{1}$ at point $F$, and $FB$ meet $AD$ at point $G$. Prove that $GM$ bisects angle $BGD$.

Let $m$ and $n$ be natural numbers. Professor Mubariz has $m$ folders and Professor Nazim has $n$ folders; initially, all folders are empty. Every day, where the day numbers are marked as $d = 1,2,3 ....,$ Prof. Mubariz is given $2018$ blue papers, and Prof. Nazim is given $2018$ orange papers. On day $d ( d = 1, 2, 3, ...),$ they both perform the following operations: [list] [*] If the $2018$ papers given to this professor are not enough to place $d$ papers in each of his folders, then he distributes all the $2018$ papers given to him to his students. If the $2018$ papers given to this professor are enough to place $d$ papers in each of his folders, firstly, he places $d$ papers in each of his folders. [*] If this professor still has papers left after the first step, he places them in the other professor's folders, with the same number in each folder and as many as possible. [*] If this professor still has papers left after the second step, he distributes them to his students. [/list] Prove that after $6$ years, the number of blue papers in one folder of Prof. Nazim will be equal to the number of orange papers in one folder of Prof. Mubariz.

A thin, uniform rod has mass $m$ and length $L$. Let the acceleration due to gravity be $g$. Let the rotational inertia of the rod about its center be $md^2$. The rod is suspended from a distance $kd$ from the center, and undergoes small oscillations with an angular frequency $\beta \sqrt{\frac{g}{d}}$. Find the maximum value of $\beta$. $ \textbf{(A)}\ 1$ $ \textbf{(B)}\ \sqrt{2}$ $ \textbf{(C)}\ 1/\sqrt{2}$ $ \textbf{(D)}\ \beta \text{ does not attain a maximum value}$ $ \textbf{(E)}\ \text{none of the above}$

Let $ABCD$ be a cyclic quadrilateral, for which $B$ and $C$ are acute angles. $M$ and $N$ are the projections of the vertex $B$ on the lines $AC$ and $AD$, respectively, $P$ and $T$ are the projections of the vertex $D$ on the lines $AB$ and $AC$ respectively, $Q$ and $S$ are the intersections of the pairs of lines $MN$ and $CD$, and $PT$ and $BC$, respectively. Prove the following statements: a) $NS \parallel PQ \parallel AC$; b) $NP=SQ$; c) $NPQS$ is a rectangle if, and only if, $AC$ is a diamteter of the circumscribed circle of quadrilateral $ABCD$.

Let $J_A$ and $J_B$ be the $A$-excenter and $B$-excenter of $\triangle ABC$. Consider a chord $\overline{PQ}$ of circle $ABC$ which is parallel to $AB$ and intersects segments $\overline{AC}$ and $\overline{BC}$. If lines $AB$ and $CP$ intersect at $R$, prove that $$\angle J_AQJ_B+\angle J_ARJ_B=180^\circ.$$

Given a positive integer $k,$ prove that for any integer $n \geq 20k,$ there exist $n - k$ pairwise distinct positive integers whose squares add up to $n(n + 1)(2n + 1)/6.$ [i]The Problem Selection Committee[/i]

Find all positive integers $n$ such that $n!$ can be written as the product of two Fibonacci numbers.

Let $x_i,$ $1 \leq i \leq n$ be real numbers with $n \geq 3.$ Let $p$ and $q$ be their symmetric sum of degree $1$ and $2$ respectively. Prove that: i) $p^2 \cdot \frac{n-1}{n}-2q \geq 0$ ii) $\left|x_i - \frac{p}{n}\right| \leq \sqrt{p^2 - \frac{2nq}{n-1}} \cdot \frac{n-1}{n}$ for every meaningful $i$.

Given the polynomial sequence $(p_{n}(x))$ satisfying $p_{1}(x)=x^{2}-1$, $p_{2}(x)=2x(x^{2}-1)$, and $p_{n+1}(x)p_{n-1}(x)=(p_{n}(x)^{2}-(x^{2}-1)^{2}$, for $n\geq 2$, let $s_{n}$ be the sum of the absolute values of the coefficients of $p_{n}(x)$. For each $n$, find a non-negative integer $k_{n}$ such that $2^{-k_{n}}s_n$ is odd.

Let $\Omega$ be the circumcircle of an acute-angled triangle $ABC$. A point $D$ is chosen on the internal bisector of $\angle ACB$ so that the points $D$ and $C$ are separated by $AB$. A circle $\omega$ centered at $D$ is tangent to the segment $AB$ at $E$. The tangents to $\omega$ through $C$ meet the segment $AB$ at $K$ and $L$, where $K$ lies on the segment $AL$. A circle $\Omega_1$ is tangent to the segments $AL, CL$, and also to $\Omega$ at point $M$. Similarly, a circle $\Omega_2$ is tangent to the segments $BK, CK$, and also to $\Omega$ at point $N$. The lines $LM$ and $KN$ meet at $P$. Prove that $\angle KCE = \angle LCP$. Poland

Let $P$ be a non-zero polynomial with non-negative real coefficients, let $N$ be a positive integer, and let $\sigma$ be a permutation of the set $\{1,2,...,n\}$. Determine the least value the sum \[\sum_{i=1}^{n}\frac{P(x_i^2)}{P(x_ix_{\sigma(i)})}\] may achieve, as $x_1,x_2,...,x_n$ run through the set of positive real numbers. [i]Fedor Petrov[/i]

Let $g(t)$ and $h(t)$ be real, continuous functions for $t\geq 0.$ Show that any function $v(t)$ satisfying the differential inequality $$\frac{dv}{dt}+g(t)v \geq h(t),\;\; v(t)=c,$$ satisfies the further inequality $v(t)\geq u(t),$ where $$\frac{du}{dt}+g(t)u = h(t),\;\; u(t)=c.$$ From this, conclude that for sufficiently small $t>0,$ the solution of $$\frac{dv}{dt}+g(t)v = v^2 ,\;\; v(t)=c$$ may be written $$v=\max_{w(t)} \left( c e^{- \int_{0}^{t} |g(s)-2w(s)| \, ds} -\int_{0}^{t} e^{-\int_{0}^{t} |g(s')-2w(s')| \, ds'} w(s)^{2} ds \right),$$ where the maximum is over all continuous functions $w(t)$ defined over some $t$-interval $[0,t_0 ].$

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows:\\ $\circ$ The first member $s_1$ is 2.\\ $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation.\\ $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation.\\ The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$. What are the 4 numbers that immediately follow $s_k = 2016$ in this sequence?

One morning, each member of Manjul’s family drank an $8$-ounce mixture of coffee and milk. The amounts of coffee and milk varied from cup to cup, but were never zero. Manjul drank $1/7$-th of the total amount of milk and $2/17$-th of the total amount of coffee. How many people are there in Manjul’s family?

Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression

In rectangle $ ABCD$, $ AB\equal{}5$ and $ BC\equal{}3$. Points $ F$ and $ G$ are on $ \overline{CD}$ so that $ DF\equal{}1$ and $ GC\equal{}2$. Lines $ AF$ and $ BG$ intersect at $ E$. Find the area of $ \triangle{AEB}$. [asy]unitsize(6mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair A=(0,0), B=(5,0), C=(5,3), D=(0,3), F=(1,3), G=(3,3); pair E=extension(A,F,B,G); draw(A--B--C--D--A--E--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",E,N); label("$F$",F,SE); label("$G$",G,SW); label("$B$",B,SE); label("1",midpoint(D--F),N); label("2",midpoint(G--C),N); label("3",midpoint(B--C),E); label("3",midpoint(A--D),W); label("5",midpoint(A--B),S);[/asy]$ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ \frac{21}{2} \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ \frac{25}{2} \qquad \textbf{(E)}\ 15$

Consider the points $P$ =$(0,0)$,$Q$ = $(1,0)$, $R$= $(2,0)$, $S$ =$(3,0)$ in the $xy$-plane. Let $A,B,C,D$ be four finite sets of colours(not necessarily distinct nor disjoint). In how many ways can $P,Q,R$ be coloured bu colours in $A,B,C$ respectively if adjacent points have to get different colours? In how many ways can $P,Q,R,S$ be coloured by colours in $A,B,C,D$ respectively if adjacent points have to get different colors?

Let $(a_i)_{i\in \mathbb{N}}$ a sequence of positive integers, such that for any finite, non-empty subset $S$ of $\mathbb{N}$, the integer$$\Pi_{k\in S} a_k -1$$is prime. Prove that the number of $a_i$'s with $i\in \mathbb{N}$ such that $a_i$ has less than $m$ distincts prime factors is finite.

An equilateral triangle $ABC$ is given. On the line through $B$ parallel to $AC$ there is a point $D$, such that $D$ and $C$ are on the same side of the line $AB$. The perpendicular bisector of $CD$ intersects the line $AB$ in $E$. Prove that triangle $CDE$ is equilateral.

In triangle $ABC$ ,$M,N,K$ are midpoints of sides $BC,AC,AB$,respectively.Construct two semicircles with diameter $AB,AC$ outside of triangle $ABC$.$MK,MN$ intersect with semicircles in $X,Y$.The tangents to semicircles at $X,Y$ intersect at point $Z$.Prove that $AZ \perp BC$.(Mehdi E'tesami Fard)

Given the pyramid $ABCD$. Let $O$ be the midpoint of edge $AC$. Given that $DO$ is the height of the pyramid, $AB=BC=2DO$ and the angle $ABC$ is right. Cut this pyramid into $8$ equal and similar to it pyramids.

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

Let $ABC$ be a triangle. Let $A_1$, $A_2$ be points on $AB$ and $AC$ respectively such that $A_1A_2 \parallel BC$ and the circumcircle of $\triangle AA_1A_2$ is tangent to $BC$ at $A_3$. Define $B_3$, $C_3$ similarly. Prove that $AA_3$, $BB_3$, and $CC_3$ are concurrent. [i]Carl Lian.[/i]

$ABC$ is a triangle, with sides $a$, $b$, $c$ , circumradius $R$, and exradii $r_a$, $r_b$, $r_c$If $2R\leq r_a$, show that $a > b$, $a > c$, $2R > r_b$, and $2R > r_c$.

Josie and Kevin are each thinking of a two digit positive integer. Josie’s number is twice as big as Kevin’s. One digit of Kevin’s number is equal to the sum of digits of Josie’s number. The other digit of Kevin’s number is equal to the difference between the digits of Josie’s number. What is the sum of Kevin and Josie’s numbers?